Fine properties of monotone maps arising in optimal transport for non-quadratic costs

The cost functions considered are $c(x,y)=h(x-y)$, where $h\in C^2({\mathbb{R}}^n)$, homogeneous of degree $p\ge 2$, with a positive definite Hessian in the unit sphere. We study multivalued monotone maps with respect to that cost and establish that they are single-valued almost everywhere. Further consequences are then deduced.